Horizon (1964–…): Season 46, Episode 12 - To Infinity and Beyond - full transcript

Zero, one, two, three, four, five...

I've seen things you
people wouldn't believe.

Things that would change
how you see this world.

Enough to drive men to madness.

DIFFERENT SPEAKERS COUNT

What is the biggest number?

Is the universe infinite?

Might every event repeat again
and again and again and again...

Your intuition is no use here.

Faith alone can't save you.

How did the universe begin?



Is the Earth just one of
uncountable copies,
tumbling through an unending void?

On one you are rich,

on another you have yet to be born.

These are the deepest
mysteries of the universe.

Ladies and gentlemen,

pray silence as I
present to you...

..infinity.

And so on.

Planet Earth is so beautiful
and so complex,

humans can barely comprehend it.

And yet humanity has always asked,
"What's over the horizon? What
lies beyond the stars?

"Is this it?"

I think infinity is one of those
things that is an essential
mystery of the universe.

What happens after
we die, why are we here,



why were we born.

Infinity is in that
class of questions and humans have
been thinking about whether

there's an end to the world
or whether the world goes on
forever,

probably since the beginning of human
thought.

It's a natural human impulse
to want to go beyond any boundary.

It is simultaneously scary and
exciting to think about.

If infinity
is real, it has implications far
beyond the world of science.

It strikes at the very heart of
what it means to be you.

There is actually, far out in space,
a planet that looks just like Earth
with people just like us.

Some will be doing
exactly the same things as we do,

even with the same names and
memories as us.

No matter how much I study the field
of cosmology and think about this,

it still makes no sense to me
that the universe is infinite.

I prefer a finite universe because
I can get my mind around that.

It's the only universe that
makes intuitive sense to me.

Infinity. Impossible to comprehend.

And yet it comes from something so
simple, a child can understand it.

One, two, three...

four, five, six...

seven, eight, nine...

10, 11, 12, 13, 14...

15, 16, 17...

18, 19, 21, 22, 23...

I was very proud first to be able
count to five, then to ten, and then

I realised that you can always keep
counting and there's no end to it.

37, 38, 39...

So I had this obvious intuition
that everybody had that

there is no end to counting
hence there must be infinity.

91, 92, 93, 94, 95, 96...

1,374, 1,375, 1,376, 1,377...

Numbers can get so vast,
it's impossible to imagine.

1,380, 1,381, 1,382...

1,2763, 1,2764, 1,2765, 1,2766...

To count to a billion, it would take
you about 30 years and to count

to a trillion is not something you
could even do in human history.

111,330, 111,331,

111,332...

1,372,365,

1,372,366...

Billion and nine, billion and ten,
billion and eleven,
billion and twelve,

For most people, I suppose the
biggest number they're likely to meet

will be somewhere in the
billions or maybe the trillions,

which might be
something like the budget deficit or

military spending or
something like that.

Mathematicians tend to use
bigger numbers than that.

Googolplexplexplex three,
googolplexplexplexplex four...

You made me do this!
Googolplexplexplex
five, googolplexplexplex six...

When I get to 199,
then that would be too hard
and I would have to stop.

TYPEWRITER KEYS CLICK

One of the largest numbers we
have a name for is a googol and
it's one followed by 100 zeros.

A hundred zeros is a lot
because each zero represents
another factor of ten.

So, it's a big number.

You might be thinking 100
zeros, isn't that many.

But a googol is far bigger than the
number of atoms in a human being,

more than the number of atoms
that make up planet Earth.

A hundred zeros is more even
than all the atoms
in the entire observable universe.

Sets your imagination
going, doesn't it?

A googol
sets your imagination going.

It knocks you off of your chair.

It's already a number that's
probably bigger than anything,
that makes sense in our experience.

But, it's a very, very tiny,
tiny, tiny large number.

A Googol itself was only a stepping
stone on the way to

a much much
larger number called a googolplex.

A googolplex is ten raised
to the power of a googol,

that is it's one followed by a
googol of zeros.

And of course it's just not
possible to imagine
the size of a number like that.

A googol has 100 zeros,
but a googolplex has so many zeros

that there's not enough space
in the entire observable universe
just to write the number down,

even if you could write each zero
on a single atom.

But from my perspective,
these are all very, very small.

One of the biggest numbers ever
used in mathematics is many times
the size of a googolplex.

It makes normal numbers
like a trillion or a billion
disappear into practically nothing.

It's a number called Graham.

Graham's number is much much
bigger than a googolplex.

In fact, it's as large
relatively to a googolplex as a
googolplex is to the number ten.

In fact, it's much much
bigger than that.

Graham's number was discovered in
the 1970s by mathematician and
former circus performer, Ron Graham.

I don't know too many other
people who have a number.

Er... It's...

It's not bad. It's not bad.

I mean... I recommend it!

Graham's number is so big,
it even made it into the
Guinness Book Of Records.

Oh, yes.

This is the 1980 edition of
Guinness Book Of World Records

and I think if we turn to page,
er, what is it, 192?

Numeration. Yes, and then numbers.

Prime numbers, perfect, highest
numbers. Here we go, highest numbers.

"The highest number
ever used in a mathematical proof

"is a bounding value published in
1977. It is known as Graham's number.

"It concerns bichromatic hypercubes
and is inexpressible without
the special arrow notation."

It's really gigantic,
I mean, it's just so large

you can't compare it with anything
you would normally associate large
numbers, like the number of atoms

in the universe or how many inches to
the furthest galaxy or something like
this, it's just way bigger than that.

So vast is Graham's number,

nobody knows how many digits it even
has, including Ron Graham himself.

In spite of the fact that
it's Graham's number, and I'm Graham,
er, I know very little about it.

I have no idea what the
first digit is.

It has one.
I have no idea what it is.

Maybe no-one will ever
know what that digit is.

Are there more zeros
than ones in the number?

Who knows?

If you look at three to the three
to the three to the n plus one...

Ron didn't just make up his number.

It's the upper limit to the solution
of a pure mathematics problem
concerning multi-dimensional cubes.

That's divisible by five,

only if the exponent three to
the three to the N plus one...

While the problem itself is
abstract, the methods Ron used to
solve it are now used to keep track

of data sent across the internet.

N plus one, minus three to the three
to the N, minus one. OK. Great. So...

Just working out the last digit

of Graham's number
is a lengthy calculation.

RON GRAHAM'S WORDS ECHO

You can't really comprehend how large
it is.

..Very large.

I don't think anybody can know
that...

But like all finite numbers,
it comes to an end...

In the last stage,
we have three to a certain power...

Eventually.

RON GRAHAM'S WORDS ECHO

So that means that the remainder,
when you divide by ten,
is always seven.

In other words, you can finally
conclude that the last digit of
Graham's number is seven.

End of the story!

Graham's number is not really
any closer to infinity
than the number one.

You didn't really get started yet,
even though you took a lot of steps

to get to Graham's number,
it takes so many more,
infinitely more to get to infinity.

Infinity is just out there.

It's just a different beast.

What's the biggest number?

Erm... 120?

Ten.

Is ten the biggest number?

Sometimes people just say, um,
seventy hundred eighty nine hundred,
but that's not even a number.

Oh, dear. Oh, that's difficult.

Well, I suppose there isn't really
one, they just go on and on and on.

There is no biggest number,

because if there were,
you could always add one to it.

TYPEWRITER KEYS CLICK

Unlike normal numbers,
infinity never comes to an end.

And that gives it some
very strange properties.

OK, so this is one of the first
things that you have to think about
if you're thinking about infinity.

You've got all the numbers - one,
two, three, four, five, six.

Let's have just a few.

And they go on forever.

Now suppose I went through and picked
out just the even numbers.

Two and four, six, eight, ten.

I'll write them down here.

Two, four, six, eight, ten.
Well, it's pretty obvious,

that there are more of
these than there are of these?

We've picked
out half of these numbers here.

Well, in fact it's not. These two
lists are exactly the same size.

And this is the first real
paradox about infinity.

There are as many even, though
there are half as many, so half of

this list has as many things in it
as the whole list does.

These sets look so different,
but they're actually the same size.

How's that possible?
Well, OK. How's that possible?

Well, you see, what are we doing,
we're counting and when you count

something you match it up with
the numbers. So if I'm counting

my sheep, for example, I count
one, two, three, four, I match
the sheep up with the numbers.

But you see, I can match the
even numbers up with the numbers.

I've already started doing it.
Here's one matched with two,
two matched with four,

three matched with six, four matched
with eight, five matched with ten,
and that goes on forever, so I've

matched all of the even numbers up
with all of the numbers, no gaps,
perfect matching between the two.

So there are the same number of these
as there are of these.

So this really is a characteristic
property of infinity and it
seems puzzling, but there it is.

This is why infinity is a bit of
a hard thing to deal with. And it
troubled people for quite some time.

Two infinite lists are
exactly the same size,

even though one appears to contain
twice as many numbers as the other.

Which is just the start.

The more mathematicians
thought about infinity,
the weirder it became.

How are you? I'm fine, thank you.

Can I check in, please?

Around 100 years ago,
one mathematician tried to explain
some of infinity's

strange properties by imagining
arriving at a hotel
with infinite rooms.

He wondered if there would
be any space for him,
even if it was fully booked?

Now in an ordinary hotel, I'd be
told, "I'm sorry we're full up.
You'll have to go somewhere else."

But in an infinite hotel,
things are rather different.
There's no problem at all.

RECEPTION BELL RINGS

The infinite hotel was dreamt up
by David Hilbert, one of the most

influential mathematicians of the
early 20th century.

Your room is upstairs.
Have a lovely stay with us. Bye-bye.

The manager can't just put me into
the last room, because there is no
last room. It's an infinite hotel.

The rooms go on forever.
There is no last room.

And all the rooms are full anyway
so, even if there were a last room,
it would have somebody in it.

But it's exactly for that reason that
it's possible to find room for me.

All you have to do is shift
the guest in room one to room two,

the guest from room two to room three
and so on down the line.

Because there's no last room,
every guest has a next room to go to

and that frees up room one,
so I can stay in room one.

So everything's fine.

It turns out,
even if the hotel was packed to the
rafters, a room can always be found.

Infinity plus one is infinity.

RECEPTION BELL RINGS

Good morning, sir. How are you?
I'm very well, thank you.

Can I check in, please?

You can see that if two guests came
in, infinity plus two is infinity.

But suppose
I came along to this hotel

with infinitely many of my friends
and we all wanted to stay and the
hotel was full, how could we do it?

What we could do is arrange that the
guests in room one moved into
room two

and the guest in room
two moved into room four,

each guest moved into the
room with double the number.

So all the even-numbered rooms
have now got people in them and all
the odd numbered rooms are now free,

so me and my friends can all stay
in the odd-numbered rooms.

So this shows that infinity
plus infinity is still infinity.

And it would
make sense that the same rules
also apply to subtraction.

But if you think infinity will
stay the same whatever you
add or subtract, then think again.

Suppose that in the morning,
all the guests left.

The number of occupied rooms would
be infinity minus infinity,
but it would be zero.

So infinity minus infinity
could be zero.

Or it could be one.

If I stayed on and all the other
guests left, infinity minus
infinity would be one in that case.

So there's no definite answer.

That's why you have to be
very careful dealing with infinity.

It's a very slippery concept.

What if I and
two other people stayed, say?

Then infinity minus
infinity would be three.

So it could be anything you like.

RECEPTION BELL RINGS

Good morning, sir. Good morning.
How are you? I'm fine, thank you.

Can I help you? Yes.
Can I check in, please?

HE LAUGHS

Oh, God!

HE LAUGHS

This is one of the reasons why you
have to very careful dealing with
a slippery character like infinity.

Not to be trusted.

Is infinity real?

Well, not to most people,
but you can't do mathematics
without infinity.

So if you don't feel comfortable
with that then you're probably not
going to become a mathematician.

Get any mathematician
to tell you about what he or
she is doing at the moment,

and their imagination will somehow
be full of this idea of infinity.

Infinity is like a landscape
in which you work.

a place in which you do mathematics.

It's not a real place.
You can't actually go there,
except in your imagination.

But to those who do mathematics,
it seems very real indeed.

But you see one of the
problems with infinity

is that it has some paradoxical
properties and very basic questions
about infinity that we can't answer.

So you do have to be
a little careful.

It's like having a
polar bear as a pet,

you've grown up together,

he's a wonderful pet,
he's big, he's fast, he plays
in the snow beautifully, but

there's always the chance that one
day he'll get annoyed with you

and bite off your head.

So, we are playing
with fire, I think.

The paradoxes associated
with infinity make some
mathematicians uncomfortable.

Not least
Professor Doron Zeilberger.

I first came across infinity like
everybody else in a very early
childhood when you start counting.

First you count to three, then to
four, then to five then to ten, then

to 100 and eventually you realise
that you can keep counting forever.

Hence there is an infinity.

341, 342, 343...
63,789, 63,790, 63...

etc, etc, ad infinitum.

But I don't think I ever liked it.

I always found something
repulsive about it.

TYPEWRITER KEYS CLICK

I prefer finite mathematics
much more to infinite mathematics.

I think that it's much more natural
much more appealing

and the
theory is much more beautiful.

It is very concrete. It's something
you can touch, something you can
feel, something you can relate to.

Infinite mathematics, to me,
is meaningless because it's
like abstract nonsense.

In my opinion, infinity is
only a fiction of the human mind.

But not believing in infinity leaves
Professor Zeilberger with a problem.

If the numbers don't go on forever,
where do they end?

When you start counting,

you seemingly can go forever,

but eventually you will reach
the biggest number and then when you
add one to it you go back to zero.

You go back to zero? Yeah.

How is that possible?

How is it not possible?
Have you ever been there?

The biggest number is much bigger
than anybody can ever think of.

It's bigger than a googol,
bigger than a googolplex,

bigger than a googolplex
to the power of a googolplex.

It's so big,
we can never envision it.

Nevertheless there is a biggest
number and if you keep counting after
that big number, we get back to zero.

Like when we walk around our planet,
if you keep walking, eventually we
get back to the place we started.

And if you think that this is
ridiculous.

Look at the alternative. It's
less ridiculous than infinity and
all the paradoxes that go with it.

Infinity may or may not exist,
God may or may not exist

but in mathematics
there should not be any place
for neither infinity nor God.

Doron isn't the first person to feel
that the infinite is an illusion.

Until recently, infinity was too
wild to be tamed by mathematics.

Too unpredictable to be used in
equations.

Aristotle believed counting
could go on forever
and that the universe was eternal.

But he refused to accept that the
universe was infinite in size.

He believed the Earth
was at the centre of the universe.

However, without an end,
there can be no middle

so he banned infinity
from his mathematics.

Infinity was discussed
by philosophers and priests
rather than mere mathematicians.

The infinite was something
closer to a god than a number.

In 1600, philosopher Giordano Bruno
claimed not only that the universe

was infinite, but there would be
many other Earths orbiting stars
just like our own sun.

His beliefs went down so poorly
with the Catholic Church,
they had him burnt at the stake.

Only God himself,
could be truly infinite.

HE SIGHS

In the mid 19th century, one man,
Gregor Cantor, made infinity
truly part of mathematics.

Something that could be used in
equations as if it were a number.

Cantor's idea was that we
can gather together maybe even

infinitely many things
into a single set,

like putting them into a bag and
think of it as just a single object.

And once it is a single object,
we can then put it into mathematics,

do calculations with it,
because it's just one object.

We don't have to know that there are
infinitely many things in the bag,
it's just a single object for us.

But making infinity
part of mathematics
produced one surprising result.

Some infinities are bigger
than others.

Cantor's great initial discovery
was that the infinity of the decimal
numbers

was larger than the
infinity of the counting numbers.

And he did this by what's now
called Cantor's diagonal argument.

So what you have to show is that
in any list of decimal numbers,

there's a decimal number
that's not on that list.

And what we can do is go
down the diagonal of this list,

so we

might take the nine here,
and the one here and the one here.

We're going down the diagonal
and we're generating
another decimal number.

Nine, one, one and so forth.

Now we take this
number and we change it.

We change the nine to an eight,
say, and we can change the ones
to twos and so forth.

And now we've generated a decimal
number, which can't be on this list.

It can't be the first one,
because the nine has been
changed to an eight.

It can't be the second one, because
in the second position, it's got a
two instead of a one and so forth.

And all the way down the list,
this number will be different from
the decimal number on the list.

So what does that mean?

It means there can be no matching
of all the decimal numbers
by the counting numbers

and that tells us that the infinity
of the decimal numbers is larger than
the infinity of the counting numbers.

There's no largest infinite number.
For every infinite number,
there's a bigger one.

There are infinities beyond
infinities and that's what we study.

In making infinity part of
mathematics, Cantor had uncovered

a whole universe of infinities, each
infinitely bigger than the last.

And that wasn't easy
for many to accept.

At the time,
I think that was a big surprise.

No-one had really thought carefully
about whether infinities could come

in different sizes, after all
infinity is infinity. How can you
have different sizes of infinity?

Cantor's breakthrough
came at a price.

Cantor ended his days in an asylum.

Was it infinity that drove him
there? Who knows? Who can tell?

He faced a lot of opposition
from his colleagues and

it was possibly that, more than
thinking about infinity itself
that was the trouble.

It was only later that mathematicians
accepted and welcomed his theories
into the body of mathematics.

Cantor's work is absolutely
fundamental to everything we do.

Today, Cantor's infinities are
part of mainstream mathematics.

And the truth is, even those who
would rather infinity didn't exist,
use it in their equations everyday.

Infinity is simpler
and quicker to manipulate
than large finite numbers.

Most mathematicians have made
an uneasy peace with infinity

and accepted it as
part of their universe.

Some have devoted
their careers to studying it.

To the person who wants to deny
infinity and say it doesn't exist,

I don't see how that
view enriches their world.

I feel sorry for them.

I mean, infinity, maybe
it doesn't exist, but
it is a beautiful subject.

I could say the stars don't exist
and stay inside, or always look down,

but then I don't see
the beauty of the stars.

And until one has a real reason to
doubt the existence of mathematical
infinity, I just don't see the point.

There's whole world of infinities
out there.

Maybe they're real,
maybe they're not.

But could infinity be part
of the world you call real?

As yet unseen through any microscope
and undetected by any telescope,

might the heavens genuinely be
unbounded and the depths of space
deeper than love itself?

Is space infinitely big
or simply unimaginably big?

TYPEWRITER KEYS CLICK

The sky never ends, so I think space
never ends, because there isn't like

a wall all around our...
all around our whole planet.

I actually think the universe
IS infinite and if I had to put odds
on it,

I would say there's
a 95% chance that it is infinite.

I think space is very, very,
very, very, very, very big.

I would say 1,000 metres.

That big.

I think the universe is infinite
on Mondays, Wednesdays and Fridays

and it's finite
the rest of the week.

I'm having a very, very
hard time making my mind up.

Space. It hasn't got
laws or electricity.

It's just like a sky.

It won't finish.

It will keep on going.

And going and it like...never stops.

The fundamental issue that most
people come up when you say the
universe may be finite, is simply

what's outside of it?

What happens when
you come to the edge?

Can't you go beyond it?

And so this leads some people
to the conclusion that the
universe has to be infinite.

It might seem obvious
that space goes on forever.

Innocent even.

But unleash infinity into the
universe and all bets are off.

He makes the extraordinary mundane
and the unbelievable, inevitable.

He makes the extraordinary mundane
and the unbelievable, inevitable.

In an infinite universe, anything
that's possible has to happen.

Even something as unlikely
as a monkey typing the
complete works of Shakespeare.

TYPEWRITER KEYS CLICK

"My bounty is as
boundless as the sea.

"My love is deep. The more I
give to thee, the more I have.

"For both are infinite."

If we imagine this monkey,
all it's doing is thumping away at
the keys completely at random.

The monkeys don't have to evolve,
they don't have to be able
to read Shakespeare, they do have

to be able to carry on typing, but
that's all, just typing at random.

"I could be bounded in a nut shell
and count myself a king of
infinite space,

"were it not
that I have bad dreams."

To test the infinite monkey theorem,

a computer was
placed in the enclosure
of a Cambridge University professor.

Typing out the complete works
of Shakespeare at random is
really going to be a big job.

It's a thick book. There's 37 plays
in here, all the poems and sonnets,

there's 884,429 words,
every word has to be in exactly

the right place, every character
has to be exactly in sequence
including the spaces in between.

And so doing that at random,
bashing away on a keyboard
is a difficult thing to do.

For the last week,
David Spiegelhalter's computer has
been randomly generating letters.

We're only generating lower case at
the moment we're not using capitals

so we're giving it a bit
of a chance like that.

And it's generating them at the
rate of 50 characters a second.

And, as you can see, it's keeping
on finding matches all the time.

If it finds a match of four letters,
four characters, it adds another
character on, a random character,

and sees if it's found a match for
five characters and so on and so on.

The programme's been running for
more than a week now
and in that time,

it's managed to generate
more than 34 million characters.

If we assume that this monkey can
type one character a second, it would

have taken 34 million seconds,
which is just over
a year's typing for our monkey,

but we've got to be kind and
give it some breaks, so I would say
just over two years probably typing.

The longest match so far
is eight letters and here's the
string, "We space lover."

This occurs once in the complete
works. It occurs in Love's Labour's
Lost, Act Two, Scene One.

So, it's in here...

Somewhere...

I've no idea where!

It's not even in
alphabetical order.

Ah, Love's Labour's Lost, here
we are. Act Two, Scene One.

Yep, I've got it.

Yeah, there we are,

Boyet says, "With that which
we lovers entitle affected."

So the actual word is "we lovers"

and that's where "we lover"
fits into it.

But eight letters
doesn't seem like very much.

No, it doesn't,
but you have to think of just

how unlikely it is to generate
the exact works of Shakespeare.

So we did some calculations and
worked out if you wanted
17 characters,

which is, "To be or not
to b..." Not even the whole, the
whole phrase. We'd have had to set

this going at about the time of the
big bang around 14 billion years ago.

And that's to get
something just twice that length.

But remember,
we've got to get five million
characters, all in the right order.

We can calculate the chance of
this happening as one in
a very large number.

It's ten with about nine
million zeros written after it.

So that's an incredibly tiny
probability very, very small. It's an
unbelievably unlikely thing to occur.

So if you imagine the
current National Lottery,

it would be like someone winning
every single time, time and again,

every single week for year
after year, for 29,000 years.

Same person.

The same person
buying their ticket and winning every
week for 29,000 years.

But if we have an infinite
amount of time, we can be
certain that it will happen.

And not just once, it's going to
happen again and again and again.

Because infinity is such a long time
that everything, no matter how
unlikely,

as long as it's possible, will occur.

Infinity is so vast, one monkey,
randomly typing forever
could easily get the job done.

TYPEWRITER KEYS CLICK

If he had an infinite amount of
time, the monkey would produce
far more than Shakespeare.

He'd produce every book
that's ever been written.

Everything from the
telephone directory to the
latest celebrity autobiography.

TYPEWRITER KEYS CLICK

But in an infinite universe,
there will be infinite number of
monkeys.

And that means, somewhere,
one of them is typing
Shakespeare right now.

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If the universe were infinite,
it seems fairly simple and benign,

but it has some really
strange consequences.

if we look
far enough away there would be
regions like the one that we're in,

there would be a room out there like
the one that we're sitting in now.

There would be Earths out
there just like ours,

except maybe
the Roman empire would still exist

or Germany would have won the war
and on a personal level out there

right now would be copies of Anthony
giving interviews

in a pink jumpsuit
or rotting in jail or filthy rich.

There would be every possible
combination. Every way your life
could have gone,

there is somebody else just like you
leading that life.

In fact, anything that we
can really conceive of

anything that's physically possible
will happen, not only that it will
happen an infinite number of times.

They're all out
there in this infinite universe.

So whether you look at this
as a good thing or a bad thing

depends a little bit on how
good you life is right now,
but there really out there.

Which is crazy. It means that there
is actually far out in space,
a planet that looks just like Earth,

with people just like us,
some will be doing
exactly the same things as we do,

even with the same
names and memories as us.

It feels a little bit spooky
to know that there are all these
other copies of me...

..but it takes a little bit of the
pressure off getting things

right all the time,

knowing that when I screw up,

one of the
other Maxes perhaps fared better.

Infinite space has consequences
it's impossible to comprehend.

Infinitely many copies of you,
identical in every possible way.

Every molecule, every heartbeat
every atom, every breath,
every thought, the same.

Each one convinced
that they're the real you.

In an infinite universe you're not
unique, you're insignificant,
you're nothing.

And, it turns out it's a relatively
simple calculation to work out

how far you would need to travel
to meet your nearest doppelganger.

Imagine a ridiculously simple
universe which only has space for

four particles and only two kinds
of particles, purple and yellow.

Then there are only 16 ways
this universe can be arranged

two times two times two times two,
16 possible arrangements.

Yellow, purple...

Purple, purple...

Almost done.

The top ones are all purple.

This means that if we arrange
a 17th universe in some random way,

like yellow, purple, purple, yellow,
it has to be a copy of one of the
existing universes. Let's see...

This one.

And this is true no matter how
we arrange these. If we do this,

then it's a copy of...this.

In other words,
this guarantees that the new
universe here is a duplicate.

And it's also easy to see that

the distance from any one to
it's nearest copy would be about
the size of this square.

In our observable universe,
there's obviously more than 16 ways
to arrange all the particles, but it

still comes out to be a finite
number, so we can use basically
the same calculation to figure out

how far away we have to go
to find an exact copy of Earth and
an exact copy of me.

All you need to do
is work out how many

subatomic particles it's possible to
cram into the observable universe.

Calculate the number of possible
configurations of those particles

and multiply that by the diameter
of that observable universe.

On the pool table
there were four balls,

so there are two to the power four
equals 16 possible arrangements.

In our actual observable universe,
we can put in up to ten to the 118
particles.

That's a huge number, but to get
the number of ways in which they can

be arranged in our universe, we have
to take about two to the power that.

So two to the power ten to the power
118. A honking big number.

Then to get to the nearest copy
of our universe, we multiply by

the size of our universe,
which is ten to the power

26 metres.

10 to the 26 is tiny
compared to this number here.

so the bottom line is that if we go
two to the power 10 to the power
118 metres away or so,

we're going to find a perfect copy
of our entire universe, of
Earth and of me.

I find this quite dizzying, frankly.

Two to the power ten to the power
118 metres is further than any human

could ever travel, but if the
universe is truly infinite,

these exact replicas of your
universe have to exist.

While no-one likes the idea
of space coming to an end,

the consequences of an infinite
universe are even more
bewildering.

No matter how much I study the field
of cosmology and think about this,

it still makes no sense to me

that the universe is infinite
and always has been infinite.

I don't understand that.

I don't pretend to understand that.

The idea that there may be an
infinite number of Earths, an
infinite number of people

having this exact talk that
I'm having right now, that just,
that doesn't compute in my brain.

I prefer a finite universe because
I can get my mind around that.

It's the only universe that
makes intuitive sense to me.

Many physicists
believe space could be curved or
even folded back on itself.

In the same way, you could
sail round the Earth forever,
if you kept on going in a straight

line through space, and could travel
long and fast enough, then you will
arrive back where you started.

You don't need infinity to
produce a universe that has no edge.

We're probably never going
to know whether our universe
is infinite or finite.

It's something
I'd really like to know.

People have wondered
about it for millennia.

The best we can say right now
is the universe is
extraordinarily large.

But cosmologists might finally be
on the verge of an answer to the
question of

whether the universe IS infinite.

And the clue that's led them there
comes from something
that has been part of your lives

since the first moment
you opened your eyes.

Light.

Light travels extraordinarily
fast, but not infinitely fast.

It travels about 300,000 kilometres
every second.

And the Moon happens to be
about 300,000 kilometres away.

So it takes light about one second
to make it from the Moon
to the Earth.

You might say the Moon is one
light second distant from the Earth.

As you gaze out into space,
you are looking back in time.

You see the Moon as it
was a second ago,

Jupiter as it was an hour ago
and your nearest galaxy,
2.5 million years in the past.

And some things are just so
far away, their light would take

longer than the age of the
universe to reach the Earth.

The most distant light ever detected

is also the oldest.

It began its journey just
400,000 years after the big bang.

13.7 billion years ago,
the universe was born.

13.7 billion years.

To infinity, it's nothing.

Seems like yesterday.

According to Big Bang theory,

after a second
the universe is ten thousand million
degrees and the first atomic nuclei

condense out of the fireball.

And darkness was on
the face of the deep...

Around 400,000 years later,
the first atoms form and light

is released into the universe.

And God called the light day.

And darkness he called night.

In the evening and the morning...

That light is still with you today.

It's called the cosmic
microwave background.

Eventually this will
become Shakespeare, Monkeys

and even you.

..we may know the traitors and the
truth...

..passing through nature...

The cosmic microwave background
is a snapshot of the universe
when it was just a baby.

Over billions of years, the
cooler, denser blue regions in this
image

will collapse to produce stars and
galaxies.

It is astonishing that just this
little picture can tell us something
potentially about infinity.

To think that we might get that sort
of insight into the universe
is pretty spectacular.

This image of the early cosmos
might now reveal whether

the universe is infinite, because
hidden within it lies a mystery,

something the big bang model alone
couldn't answer.

What we see when
we're looking at this

map of the microwave
background radiation

is that it's showing
us what temperature

this radiation has all over the sky.

So this is an image
of the whole sky.

If you look at this red splotch
versus that blue splotch there's
only a difference of about

a ten thousandth of a degree so in
fact, this microwave background
radiation is incredibly uniform.

Now for those two to be so similar,
it seems that some sort of physical
process,

some sort of agreement
should have taken place and this
is a rather baffling mystery.

How did they do it,
how did they come to this agreement
in their temperature?

This was a real enigma
for the standard big bang cosmology.
It was a real puzzle.

In the big bang model,

there was no way to explain how
distant parts of the universe could
have such similar temperatures.

To make sense of it,
physicists needed something else.

A new theory
of the early universe.

A theory they called inflation.

HE INFLATES GLOBE

Inflation says the early universe
expanded much faster
and further than previously thought,

a million, million, million
times, in less than

a billionth of a billionth of
a second.

If I faint, I'll sue you!

This explains the uniform
temperature of the cosmic microwave
background because

everything you see was stretched out
from a small and uniform
part of the whole universe.

Now just imagine I did all this in
ten to the power minus 32 seconds.

That would be inflation.

Inflation was devised to explain the
finite observable universe

and it does a very nice job of doing
that, but it has a sort of

side effect or a very interesting
property that once you get inflation
started, it just keeps going.

It takes on a life of its
own, like a genie you've
let out of the bottle.

The theory of cosmological inflation

actually produces
an infinite universe.

Inflation predicts your
universe never stops expanding

and may, in fact, be infinite.

But following the mathematics to
its logical conclusion predicts
an even more disturbing outcome.

Your infinite universe might not
be the only one.

We used to think
that inflation only gave us one big
bang and one infinite space, but now

it's becoming clear that actually it
never stops and instead gives us an
infinite number of infinite spaces.

There are multiple universes,
infinitely many multiple universes,

infinitely many
infinite universes even.

Inflation was devised to explain
the finite observable universe, and
it does a very nice job of that...

WORDS ECHO

In an infinite universe, anything
possible happens all the time.

But with infinite universes, the
impossible is happening right now.

Because in some of those universes,
the laws of physics that govern
your world simply don't apply.

What isn't appreciated by many even
in the physics community is that
this model of these infinitely

many infinite universes is actually
probably our current best bet as to
what the real universe looks like.

It's baffling and it's
mind bending but that's where our
road of cosmology has taken us,

to this confrontation with real
infinity.

I think we should expect us not
to be able to intuitively grasp
the ultimate nature of space

and everything because we have
intuition only for the things which
were useful for our ancestors.

And we shouldn't expect
our intuition to work for really

big questions about the
ultimate nature of reality.

If one of our ancestors spent
too much time thinking about

what's outside space, you know,
they wouldn't have noticed that
there was a tiger sneaking up

from behind and they would have been
cleaned right out of the gene pool.

So it's very important for
us scientists to not diss
ideas just because they feel weird.

Fortunately, our math doesn't
have any inhibitions and we could
still

calculate all these things
even if they seem completely
counter intuitive and it's only

through the math that we're able to
actually deal with all these ideas.

Counting has led you to an infinite
mathematical world of infinities,

each
infinitely larger than the last.

And gazing out into the
furthest depths of space,

some see an infinite universe,
itself just one of infinitely many.

Infinity is a big topic I
don't think it's gonna be understood
fully in any finite period of time.

We have a hint of just how rich that
realm is, but we haven't understood
the smallest fraction of it.

This, of course, is because, by its
very nature, the subject of infinity
is a vast and infinite subject.

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